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As noted in Carlo Beenakker's comment, your inequality is a direct application of Theorem 3.5 in the linked paper: in that theorem, take $d_j=X_j$, $r=\sqrt{2M^2 T\ln(2/\delta)}$, $b_*^2=M^2T$, and $D …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\th …

The independence will be then in general lost. E.g., let $X_1,\dots,X_n$ be independent random variables each uniformly distributed on $[0,1]$. Let $M:=\max(X_1,\dots,X_n)=X_\nu$, so that $\nu$ is uni …

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\la …

$\newcommand\R{\mathbb R}$In Theorem 2.3 of Buja, Logan, Reeds, and Shepp, it was shown by a modification of a method of Lévy (Lemma 2.2 by Buja etal, with a proof like that in another answer) that t …

Your question is Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary? It is never possible to define physical phenomena directly in mathematical l …

$\renewcommand{\r}{\mathbf r}$This surface, say $S$, is naturally parametrized as follows: \begin{equation*} [0,2]\times[0,2\pi]\ni(y,t)\mapsto\r(y,t):=(x,y,z)=(r(y)\cos t,y,r(y)\sin t)\in S, \end{equ …

You may want to try Combinatorics, Probability & Computing or The Electronic Journal of Combinatorics. Your chances at getting published depend on the quality of your paper.

As written in my paper [1], the inequality $$P(f^*>r) \le 2\exp\big(-r^2/2(p-1)\big) $$ in Theorem 3 in [1] for martingales in $\mathcal{X}=L^p$ can be compared with the inequality $$ P(f^*>r) \le …

$\newcommand{\si}{\sigma}$ Let us prove a stronger estimate of $EY$, and let us do that under less restrictive conditions. Namely, let us prove that \begin{equation*} EY-\sqrt n=O(1/\sqrt n) \tag{1} …

$\newcommand{\ep}{\varepsilon} $ Let $X$ be any nonnegative random variable (r.v.) with finite mean $\mu>0$ and variance $\sigma^2<\infty$. For any real $u>0$, we have $\ln\frac xu\le\frac xu-1$ for a …

Let $S$ be the survival time. Then $$P(S\ge s)=\binom{n-s}a\Big/\binom na$$ for $s=0,1,\dots$. So, $$ES=-1+\sum_{s=0}^\infty P(S\ge s)=\frac{n+1}{a+1}-1.$$ So, $n=3$, $a=1$, $n^*=8$, $a^*=3$ will do. …

Let $X_1,X_2,\dots$ be iid random vectors each uniformly distributed on $S^{d-1}$. Let $S_n:=\sum_1^n X_i$. By the symmetry, $EX_1=0$. Also, $1=|X_1|^2=\sum_{j=1}^d X_{1j}^2$, where $X_1=(X_{11},\dots …

Let $r:=\rho$, $S(r):=S$, and $a:=1/6$. Let us show that \begin{equation*} S(r)< Cr(1-r)^{a-1}\quad\text{and}\quad S(r)\sim C(1-r)^{a-1},\quad\text{where}\quad C:=\Gamma(1-a). \tag{0} \end{equation …

The answer is no. Indeed, if a martingale is a.s. everywhere differentiable, then its quadratic variation is a.s $0$. So, by the Burkholder--Davis--Gundy inequality, the martingale is a.s. constant. …